On Relative Length of Long Paths and Cycles in Graphs

TL;DR

This paper improves a classical Hamiltonian graph condition by replacing a lower bound involving the number of vertices with one involving the length of the longest path, refining the criteria for Hamiltonicity.

Contribution

The paper advances graph theory by strengthening a known Hamiltonian condition, replacing the bound n+κ with p+κ, where p is the length of the longest path.

Findings

Improved Hamiltonian condition with p+κ bound

Reduced the lower bound requirement from n+κ to p+κ

Enhanced understanding of path and cycle lengths in graphs

Abstract

Let $G$ be a graph on $n$ vertices, $p$ the order of a longest path and $\kappa$ the connectivity of $G$. In 1989, Bauer, Broersma Li and Veldman proved that if $G$ is a 2-connected graph with $d(x)+d(y)+d(z)\ge n+\kappa$ for all triples $x,y,z$ of independent vertices, then $G$ is hamiltonian. In this paper we improve this result by reducing the lower bound $n+\kappa$ to $p+\kappa$.